If both 5^2 and 3^3 are factors of n x (2^5) x (6^2) x (7^3)

First, if a number, lets say X, is a factor of another number, lets say Y . Then Y is divisible by X

In the question, both 5^2 and 3^3 are factors of n x (2^5) x (6^2) x (7^3), so n x (2^5) x (6^2) x (7^3) must be divisible by 5^2 and 3^3

we can rewrite n x (2^5) x (6^2) x (7^3) as n x (2^5) x (2^2) x (3^2) x (7^3)

so to make (2^5) x (2^2) x (3^2) x (7^3) divisible by 5^2 and 3^3, we need 5^2 and 3^1 ( if Y is divisible by X, then all prime factors of X must also be prime factors of Y)

so n= 5^2 and 3^1 which equals 75.

hope you find it helpful.

Just compare 5^2 and 3^3 with >>>>>>>> (2^5) x (6^2) x (7^3)

5^2 = 25 is not present; also 3^2 is present instead of 3^3, so one additional 3 is required as well

So, n= 25x3 = 75 = Answer = D

Given \(n(2^5)(6^2)(7^3)\) is divisible by both \(5^2\) and \(3^3\).

Therefore \(n(2^5)(6^2)(7^3)\) will be divisible by LCM of \(5^2\) and \(3^3\).

LCM of \(5^2\) and \(3^3 = (5^2)(3^3)\)

By Simplifying the expression we get;

\(\frac{n(2^5)(6^2)(7^3)}{(5^2)(3^3)}\)

\(\frac{n(2^5)(2*3)^2(7^3)}{(5^2)(3^3)}\)

\(\frac{n(2^5)(2^2)(3^2)(7^3)}{(5^2)(3^3)}\)

\(\frac{n(2^7)(3^2)(7^3)}{(5^2)(3^3)}\)

\(\frac{n(2^7)(7^3)}{(5^2)(3)}\)

\(2\) and \(7\) are not divisible by \(5\) or \(3\). Hence \(n\) should be a multiple divisible by \((5^2*3)\)

Therefor \(n(2^7)(7^3)\) to be divisible by \((5^2*3)\), smallest possible of \(n\) Should be \(= 5^2*3 = 25*3 = 75\)

Answer (D)...

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Solution

Given:

• \(5^2\) and 3^3 are factors of \(n * (2^5) * (6^2) * (7^3)\)

To find:
• Smallest possible positive value of n.

Approach and Working:

• \(n * (2^5) * (6^2) * (7^3) = 2^5 * (2^2*3^2) *7^3 *n\)
• = \(2^7 * 3^2 *7^3 *n\)
• \(5^2\) and \(3^3\)and both factors of \(2^7 * 3^2 *7^3 *n\).
o Hence, \(2^7 * 3^2 *7^3 *n\) are divisible by \(5^2 * 3^3\).
o Thus, n= \(3* 5^2\) = 75

Hence, the correct answer is option D.
Answer: D

5^2 = 25 is not present; also 3^2 is present instead of 3^3, so one additional 3 is required as well

so, multiply 25 with 3 =75 will be answer

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